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SQL management studio allows to create scripts for all db objects, however I so far couldn't find a way to correctly script a schema or user. The permissions of a user on a schema are not included in the script that is created. Did I make something wrong or is MSFT her a bit sloppy ?

You will need to script permissions in two steps, roles and objects. As Kin alludes to, you can use sys.database_permissions for the objects, but you will want to use sys.database_principals and sys.database_role_members for the role membership. The following SQL will work for SQL 2012 only (previous versions should use sp_addrolemember for roles) and should be executed in the context of the database you are scripting permissions for. Filter on a specific user as necessary.